This work considers a theoretical model for quasi-spherical subsonic accretion onto slowly rotating magnetized neutron stars. In this regime the accreting matter settles down subsonically onto the rotating magnetosphere, forming an extended quasi-static shell. The shell mediates the angular momentum transfer to/from the rotating neutron star magnetosphere by large-scale convective motions, which for observed pulsars lead to an almost so-angular-momentum rotation law with ω ∼ 1/R 2 inside the shell. The accretion rate through the shell is determined by the ability of the plasma to enter the magnetosphere due to Rayleigh-Taylor instabilities while taking cooling into account. The settling regime of accretion is possible for moderate accretion ratesṀ Ṁ * ≃ 4 × 10 16 g/s. At higher accretion rates a free-fall gap above the neutron star magnetosphere appears due to rapid Compton cooling, and accretion becomes highly non-stationary. From observations of spin-up/spin-down rates of quasi-spherically wind accreting equilibrium X-ray pulsars with known orbital periods (like e.g. GX 301-2 and Vela X-1), it is possible to determine the main dimensionless parameters of the model, as well as to estimate the magnetic field on the surface of the neutron star. For equilibrium pulsars with independent measurements of the magnetic field, the model also allows us to estimate the velocity of the stellar wind from the companion without the use of complicated spectroscopic measurements. For non-equilibrium pulsars, it can be shown that there exists a maximum possible value of the spin-down rate of the accreting neutron star. From observations of the spin-down rate and the X-ray luminosity in such pulsars (e. g. GX 1+4, SXP 1062 and 4U 2206+54) we are able to estimate a lower limit on the neutron star magnetic field, which in all exemplified cases turns out to be close to the standard one and in agreement with cyclotron line measurements. The model further explains both the spin-up/spin-down of the pulsar frequency on large time-scales and the irregular short-term frequency fluctuations, which may correlate or anti-correlate with the X-ray luminosity fluctuations, seen in different systems.